11.5: A.5- Inner Product and Projections
-
- Last updated
- Save as PDF
Inner Product and Orthogonality
To do basic geometry, we need length, and we need angles. We have already seen the euclidean length, so let us figure out how to compute angles. Mostly, we are worried about the right angle \(^{1}\) .
Given two (column) vectors in \({\mathbb{R}}^n\), we define the (standard) inner product as the dot product: \[\langle \vec{x} , \vec{y} \rangle = \vec{x} \cdot \vec{y} = \vec{y}^T \vec{x} = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n = \sum_{i=1}^n x_i y_i . \nonumber \] Why do we seemingly give a new notation for the dot product? Because there are other possible inner products, which are not the dot product, although we will not worry about others here. An inner product can even be defined on spaces of functions as we do in Chapter 4 : \[\langle f(t) , g(t) \rangle = \int_{a}^{b} f(t) g(t) \, dt . \nonumber \] But we digress.
The inner product satisfies the following rules:
- \(\langle \vec{x} , \vec{x} \rangle \geq 0\), and \(\langle \vec{x} , \vec{x} \rangle = 0\) if and only if \(\vec{x} = 0\),
- \(\langle \vec{x} , \vec{y} \rangle = \langle \vec{y} , \vec{x} \rangle\),
- \(\langle a\vec{x} , \vec{y} \rangle = \langle \vec{x} , a\vec{y} \rangle = a \langle \vec{x} , \vec{y} \rangle\),
- \(\langle \vec{x} + \vec{y} , \vec{z} \rangle = \langle \vec{x} , \vec{z} \rangle + \langle \vec{y} , \vec{z} \rangle\) and \(\langle \vec{x}, \vec{y} + \vec{z} \rangle = \langle \vec{x} , \vec{y} \rangle + \langle \vec{x} , \vec{z} \rangle\).
Anything that satisfies the properties above can be called an inner product, although in this section we are concerned with the standard inner product in \({\mathbb{R}}^n\).
The standard inner product gives the euclidean length: \[\lVert{\vec{x}}\rVert = \sqrt{\langle \vec{x}, \vec{x} \rangle} = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} . \nonumber \] How does it give angles?
You may recall from multivariable calculus, that in two or three dimensions, the standard inner product (the dot product) gives you the angle between the vectors: \[\langle \vec{x}, \vec{y} \rangle = \lVert{\vec{x}}\rVert \lVert{\vec{y}}\rVert \cos \theta. \nonumber \] That is, \(\theta\) is the angle that \(\vec{x}\) and \(\vec{y}\) make when they are based at the same point.
In \({\mathbb{R}}^n\) (any dimension), we are simply going to say that \(\theta\) from the formula is what the angle is. This makes sense as any two vectors based at the origin lie in a 2-dimensional plane (subspace), and the formula works in 2 dimensions. In fact, one could even talk about angles between functions this way, and we do in Chapter 4 , where we talk about orthogonal functions (functions at right angle to each other).
To compute the angle we compute \[\cos \theta = \frac{\langle \vec{x}, \vec{y} \rangle}{\lVert{\vec{x}}\rVert \lVert{\vec{y}}\rVert} . \nonumber \] Our angles are always in radians. We are computing the cosine of the angle, which is really the best we can do. Given two vectors at an angle \(\theta\), we can give the angle as \(-\theta\), \(2\pi-\theta\), etc., see Figure \(\PageIndex{1}\). Fortunately, \(\cos \theta = \cos (-\theta) = \cos(2\pi - \theta)\). If we solve for \(\theta\) using the inverse cosine \(\cos^{-1}\), we can just decree that \(0 \leq \theta \leq \pi\).
Example \(\PageIndex{1}\)
Let us compute the angle between the vectors \((3,0)\) and \((1,1)\) in the plane. Compute \[\cos \theta = \frac{\bigl\langle (3,0) , (1,1) \bigr\rangle}{\lVert(3,0)\rVert \lVert(1,1)\rVert} = \frac{3 + 0}{3 \sqrt{2}} = \frac{1}{\sqrt{2}} . \nonumber \] Therefore \(\theta = \frac{\pi}{4}\).
As we said, the most important angle is the right angle. A right angle is \(\frac{\pi}{2}\) radians, and \(\cos (\frac{\pi}{2}) = 0\), so the formula is particularly easy in this case. We say vectors \(\vec{x}\) and \(\vec{y}\) are orthogonal if they are at right angles, that is if \[\langle \vec{x} , \vec{y} \rangle = 0 . \nonumber \] The vectors \((1,0,0,1)\) and \((1,2,3,-1)\) are orthogonal. So are \((1,1)\) and \((1,-1)\). However, \((1,1)\) and \((1,2)\) are not orthogonal as their inner product is \(3\) and not \(0\).
Orthogonal Projection
A typical application of linear algebra is to take a difficult problem, write everything in the right basis, and in this new basis the problem becomes simple. A particularly useful basis is an orthogonal basis, that is a basis where all the basis vectors are orthogonal. When we draw a coordinate system in two or three dimensions, we almost always draw our axes as orthogonal to each other.
Generalizing this concept to functions, it is particularly useful in Chapter 4 to express a function using a particular orthogonal basis, the Fourier series.
To express one vector in terms of an orthogonal basis, we need to first project one vector onto another. Given a nonzero vector \(\vec{v}\), we define the orthogonal projection of \(\vec{w}\) onto \(\vec{v}\) as \[\operatorname{proj}_{\vec{v}}(\vec{w}) = \left( \frac{\langle \vec{w} , \vec{v} \rangle}{ \langle \vec{v} , \vec{v} \rangle} \right) \vec{v} . \nonumber \] For the geometric idea, see Figure \(\PageIndex{2}\). That is, we find the "shadow of \(\vec{w}\)" on the line spanned by \(\vec{v}\) if the direction of the sun’s rays were exactly perpendicular to the line. Another way of thinking about it is that the tip of the arrow of \(\operatorname{proj}_{\vec{v}}(\vec{w})\) is the closest point on the line spanned by \(\vec{v}\) to the tip of the arrow of \(\vec{w}\). In terms of euclidean distance, \(\vec{u} = \operatorname{proj}_{\vec{v}}(\vec{w})\) minimizes the distance \(\lVert \vec{w} - \vec{u} \rVert\) among all vectors \(\vec{u}\) that are multiples of \(\vec{v}\). Because of this, this projection comes up often in applied mathematics in all sorts of contexts we cannot solve a problem exactly: We can’t always solve "Find \(\vec{w}\) as a multiple of \(\vec{v}\)" but \(\operatorname{proj}_{\vec{v}}(\vec{w})\) is the best "solution."
The formula follows from basic trigonometry. The length of \(\operatorname{proj}_{\vec{v}}(\vec{w})\) should be \(\cos \theta\) times the length of \(\vec{w}\), that is \((\cos \theta)\lVert\vec{w}\rVert\). We take the unit vector in the direction of \(\vec{v}\), that is, \(\frac{\vec{v}}{\lVert \vec{v} \rVert}\) and we multiply it by the length of the projection. In other words, \[\operatorname{proj}_{\vec{v}}(\vec{w}) = (\cos \theta) \lVert \vec{w} \rVert \frac{\vec{v}}{\lVert \vec{v} \rVert} = \frac{(\cos \theta) \lVert \vec{w} \rVert \lVert \vec{v} \rVert}{ {\lVert \vec{v} \rVert}^2 } \vec{v} = \frac{\langle \vec{w}, \vec{v} \rangle}{ \langle \vec{v}, \vec{v} \rangle } \vec{v} . \nonumber \]
Example \(\PageIndex{2}\)
Suppose we wish to project the vector \((3,2,1)\) onto the vector \((1,2,3)\). Compute \[\begin{align}\begin{aligned} \operatorname{proj}_{(1,2,3)} \bigl( (3,2,1) \bigr) = \frac{\langle (3,2,1) , (1,2,3) \rangle}{\langle (1,2,3) , (1,2,3) \rangle} (1,2,3) & = \frac{3 \cdot 1 + 2 \cdot 2 + 1 \cdot 3}{ 1 \cdot 1 + 2 \cdot 2 + 3 \cdot 3} (1,2,3) \\ & = \frac{10}{14} (1,2,3) = \left(\frac{5}{7},\frac{10}{7},\frac{15}{7}\right) . \end{aligned}\end{align} \nonumber \] Let us double check that the projection is orthogonal. That is \(\vec{w}-\operatorname{proj}_{\vec{v}}(\vec{w})\) ought to be orthogonal to \(\vec{v}\), see the right angle in Figure \(\PageIndex{2}\). That is, \[(3,2,1) - \operatorname{proj}_{(1,2,3)} \bigl( (3,2,1) \bigr) = \left(3-\frac{5}{7},2-\frac{10}{7},1-\frac{15}{7}\right) = \left(\frac{16}{7},\frac{4}{7},\frac{-8}{7}\right) \nonumber \] ought to be orthogonal to \((1,2,3)\). We compute the inner product and we had better get zero: \[\left\langle \left(\frac{16}{7},\frac{4}{7},\frac{-8}{7}\right) , (1,2,3) \right\rangle = \frac{16}{7} \cdot 1 + \frac{4}{7} \cdot 2 -\frac{8}{7} \cdot 3 = 0 . \nonumber \]
Orthogonal Basis
As we said, a basis \(\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\) is an orthogonal basis if all vectors in the basis are orthogonal to each other, that is, if \[\langle \vec{v}_j , \vec{v}_k \rangle = 0 \nonumber \] for all choices of \(j\) and \(k\) where \(j \not= k\) (a nonzero vector cannot be orthogonal to itself). A basis is furthermore called an orthonormal basis if all the vectors in a basis are also unit vectors, that is, if all the vectors have magnitude 1. For example, the standard basis \(\{ (1,0,0), (0,1,0), (0,0,1) \}\) is an orthonormal basis of \({\mathbb{R}}^3\): Any pair is orthogonal, and each vector is of unit magnitude.
The reason why we are interested in orthogonal (or orthonormal) bases is that they make it really simple to represent a vector (or a projection onto a subspace) in the basis. The simple formula for the orthogonal projection onto a vector gives us the coefficients. In Chapter 4 , we use the same idea by finding the correct orthogonal basis for the set of solutions of a differential equation. We are then able to find any particular solution by simply applying the orthogonal projection formula, which is just a couple of a inner products.
Let us come back to linear algebra. Suppose that we have a subspace and an orthogonal basis \(\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n\). We wish to express \(\vec{x}\) in terms of the basis. If \(\vec{x}\) is not in the span of the basis (when it is not in the given subspace), then of course it is not possible, but the following formula gives us at least the orthogonal projection onto the subspace, or in other words, the best approximation in the subspace.
First suppose that \(\vec{x}\) is in the span. Then it is the sum of the orthogonal projections: \[\vec{x} = \operatorname{proj}_{\vec{v}_1} ( \vec{x} ) + \operatorname{proj}_{\vec{v}_2} ( \vec{x} ) + \cdots + \operatorname{proj}_{\vec{v}_n} ( \vec{x} ) = \frac{\langle \vec{x}, \vec{v}_1 \rangle}{ \langle \vec{v}_1, \vec{v}_1 \rangle } \vec{v}_1 + \frac{\langle \vec{x}, \vec{v}_2 \rangle}{ \langle \vec{v}_2, \vec{v}_2 \rangle } \vec{v}_2 + \cdots + \frac{\langle \vec{x}, \vec{v}_n \rangle}{ \langle \vec{v}_n, \vec{v}_n \rangle } \vec{v}_n . \nonumber \] In other words, if we want to write \(\vec{x} = a_1 \vec{v}_1 + a_2 \vec{v}_2 + \cdots + a_n \vec{v}_n\), then \[a_1 = \frac{\langle \vec{x}, \vec{v}_1 \rangle}{ \langle \vec{v}_1, \vec{v}_1 \rangle } , \quad a_2 = \frac{\langle \vec{x}, \vec{v}_2 \rangle}{ \langle \vec{v}_2, \vec{v}_2 \rangle } , \quad \ldots , \quad a_n = \frac{\langle \vec{x}, \vec{v}_n \rangle}{ \langle \vec{v}_n, \vec{v}_n \rangle } . \nonumber \] Another way to derive this formula is to work in reverse. Suppose that \(\vec{x} = a_1 \vec{v}_1 + a_2 \vec{v}_2 + \cdots + a_n \vec{v}_n\). Take an inner product with \(\vec{v}_j\), and use the properties of the inner product: \[\begin{align}\begin{aligned} \langle \vec{x} , \vec{v}_j \rangle & = \langle a_1 \vec{v}_1 + a_2 \vec{v}_2 + \cdots + a_n \vec{v}_n , \vec{v}_j \rangle \\ & = a_1 \langle \vec{v}_1 , \vec{v}_j \rangle + a_2 \langle \vec{v}_2 , \vec{v}_j \rangle + \cdots + a_n \langle \vec{v}_n , \vec{v}_j \rangle . \end{aligned}\end{align} \nonumber \] As the basis is orthogonal, then \(\langle \vec{v}_k , \vec{v}_j \rangle = 0\) whenever \(k \not= j\). That means that only one of the terms, the \(j^{\text{th}}\) one, on the right hand side is nonzero and we get \[\langle \vec{x} , \vec{v}_j \rangle = a_j \langle \vec{v}_j , \vec{v}_j \rangle . \nonumber \] Solving for \(a_j\) we find \(a_j = \frac{\langle \vec{x}, \vec{v}_j \rangle}{ \langle \vec{v}_j, \vec{v}_j \rangle }\) as before.
Example \(\PageIndex{3}\)
The vectors \((1,1)\) and \((1,-1)\) form an orthogonal basis of \({\mathbb{R}}^2\). Suppose we wish to represent \((3,4)\) in terms of this basis, that is, we wish to find \(a_1\) and \(a_2\) such that \[(3,4) = a_1 (1,1) + a_2 (1,-1) . \nonumber \] We compute: \[a_1 = \frac{\langle (3,4), (1,1) \rangle}{ \langle (1,1), (1,1) \rangle } = \frac{7}{2}, \qquad a_2 = \frac{\langle (3,4), (1,-1) \rangle}{ \langle (1,-1), (1,-1) \rangle } = \frac{-1}{2} . \nonumber \] So \[(3,4) = \frac{7}{2} (1,1) + \frac{-1}{2} (1,-1) . \nonumber \]
If the basis is orthonormal rather than orthogonal, then all the denominators are one. It is easy to make a basis orthonormal—divide all the vectors by their size. If you want to decompose many vectors, it may be better to find an orthonormal basis. In the example above, the orthonormal basis we would thus create is \[\left( \frac{1}{\sqrt{2}} , \frac{1}{\sqrt{2}} \right) , \quad \left( \frac{1}{\sqrt{2}} , \frac{-1}{\sqrt{2}} \right) . \nonumber \] Then the computation would have been \[\begin{align}\begin{aligned} (3,4) & = \left\langle (3,4) , \left( \frac{1}{\sqrt{2}} , \frac{1}{\sqrt{2}} \right) \right\rangle \left( \frac{1}{\sqrt{2}} , \frac{1}{\sqrt{2}} \right) + \left\langle (3,4) , \left( \frac{1}{\sqrt{2}} , \frac{-1}{\sqrt{2}} \right) \right\rangle \left( \frac{1}{\sqrt{2}} , \frac{-1}{\sqrt{2}} \right) \\ & = \frac{7}{\sqrt{2}} \left( \frac{1}{\sqrt{2}} , \frac{1}{\sqrt{2}} \right) + \frac{-1}{\sqrt{2}} \left( \frac{1}{\sqrt{2}} , \frac{-1}{\sqrt{2}} \right) . \end{aligned}\end{align} \nonumber \] Maybe the example is not so awe inspiring, but given vectors in \({\mathbb{R}}^{20}\) rather than \({\mathbb{R}}^2\), then surely one would much rather do 20 inner products (or 40 if we did not have an orthonormal basis) rather than solving a system of twenty equations in twenty unknowns using row reduction of a \(20 \times 21\) matrix.
As we said above, the formula still works even if \(\vec{x}\) is not in the subspace, although then it does not get us the vector \(\vec{x}\) but its projection. More concretely, suppose that \(S\) is a subspace that is the span of \(\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\) and \(\vec{x}\) is any vector. Let \(\operatorname{proj}_{S}(\vec{x})\) be the vector in \(S\) that is the closest to \(\vec{x}\). Then \[\operatorname{proj}_{S}(\vec{x}) = \frac{\langle \vec{x}, \vec{v}_1 \rangle}{ \langle \vec{v}_1, \vec{v}_1 \rangle } \vec{v}_1 + \frac{\langle \vec{x}, \vec{v}_2 \rangle}{ \langle \vec{v}_2, \vec{v}_2 \rangle } \vec{v}_2 + \cdots + \frac{\langle \vec{x}, \vec{v}_n \rangle}{ \langle \vec{v}_n, \vec{v}_n \rangle } \vec{v}_n . \nonumber \] Of course, if \(\vec{x}\) is in \(S\), then \(\operatorname{proj}_{S}(\vec{x}) = \vec{x}\), as the closest vector in \(S\) to \(\vec{x}\) is \(\vec{x}\) itself. But true utility is obtained when \(\vec{x}\) is not in \(S\). In much of applied mathematics, we cannot find an exact solution to a problem, but we try to find the best solution out of a small subset (subspace). The partial sums of Fourier series from Chapter 4 are one example. Another example is least square approximation to fit a curve to data. Yet another example is given by the most commonly used numerical methods to solve partial differential equations, the finite element methods.
Example \(\PageIndex{4}\)
The vectors \((1,2,3)\) and \((3,0,-1)\) are orthogonal, and so they are an orthogonal basis of a subspace \(S\): \[S = \operatorname{span} \bigl\{ (1,2,3), (3,0,-1) \bigr\} . \nonumber \] Let us find the vector in \(S\) that is closest to \((2,1,0)\). That is, let us find \(\operatorname{proj}_{S}\bigl((2,1,0)\bigr)\). \[\begin{align}\begin{aligned} \operatorname{proj}_{S}\bigl((2,1,0)\bigr) & = \frac{\langle (2,1,0), (1,2,3) \rangle}{ \langle (1,2,3), (1,2,3) \rangle } (1,2,3) + \frac{\langle (2,1,0), (3,0,-1) \rangle}{ \langle (3,0,-1), (3,0,-1) \rangle } (3,0,-1) \\ & = \frac{2}{7} (1,2,3) + \frac{3}{5} (3,0,-1) \\ &= \left( \frac{73}{35} , \frac{4}{7} , \frac{9}{35} \right) . \end{aligned}\end{align} \nonumber \]
Gram–Schmidt Process
Before leaving orthogonal bases, let us note a procedure for manufacturing them out of any old basis. It may not be difficult to come up with an orthogonal basis for a 2-dimensional subspace, but for a 20-dimensional subspace, it seems a daunting task. Fortunately, the orthogonal projection can be used to "project away" the bits of the vectors that are making them not orthogonal. It is called the Gram-Schmidt process .
We start with a basis of vectors \(\vec{v}_1,\vec{v}_2, \ldots, \vec{v}_n\). We construct an orthogonal basis \(\vec{w}_1, \vec{w}_2, \ldots, \vec{w}_n\) as follows. \[\begin{align}\begin{aligned} \vec{w}_1 & = \vec{v}_1 , \\ \vec{w}_2 & = \vec{v}_2 - \operatorname{proj}_{\vec{w}_1}(\vec{v}_2) , \\ \vec{w}_3 & = \vec{v}_3 - \operatorname{proj}_{\vec{w}_1}(\vec{v}_3) - \operatorname{proj}_{\vec{w}_2}(\vec{v}_3) , \\ \vec{w}_4 & = \vec{v}_4 - \operatorname{proj}_{\vec{w}_1}(\vec{v}_4) - \operatorname{proj}_{\vec{w}_2}(\vec{v}_4) - \operatorname{proj}_{\vec{w}_3}(\vec{v}_4) , \\ & \vdots \\ \vec{w}_n & = \vec{v}_n - \operatorname{proj}_{\vec{w}_1}(\vec{v}_n) - \operatorname{proj}_{\vec{w}_2}(\vec{v}_n) - \cdots - \operatorname{proj}_{\vec{w}_{n-1}}(\vec{v}_n) .\end{aligned}\end{align} \nonumber \] What we do is at the \(k^{\text{th}}\) step, we take \(\vec{v}_k\) and we subtract the projection of \(\vec{v}_k\) to the subspace spanned by \(\vec{w}_1,\vec{w}_2,\ldots,\vec{w}_{k-1}\).
Example \(\PageIndex{5}\)
Consider the vectors \((1,2,-1)\), and \((0,5,-2)\) and call \(S\) the span of the two vectors. Let us find an orthogonal basis of \(S\): \[\begin{align}\begin{aligned} \vec{w}_1 & = (1,2,-1) , \\ \vec{w}_2 & = (0,5,-2) - \operatorname{proj}_{(1,2,-1)}\bigl((0,2,-2)\bigr) \\ & = (0,1,-1) - \frac{\langle (0,5,-2), (1,2,-1) \rangle}{ \langle (1,2,-1), (1,2,-1) \rangle } (1,2,-1) = (0,5,-2) - 2 (1,2,-1) = (-2,1,0) .\end{aligned}\end{align} \nonumber \] So \((1,2,-1)\) and \((-2,1,0)\) span \(S\) and are orthogonal. Let us check: \(\langle (1,2,-1) , (-2,1,0) \rangle = 0\).
Suppose we wish to find an orthonormal basis, not just an orthogonal one. Well, we simply make the vectors into unit vectors by dividing them by their magnitude. The two vectors making up the orthonormal basis of \(S\) are: \[\frac{1}{\sqrt{6}} (1,2,-1) = \left( \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}, \frac{-1}{\sqrt{6}} \right) , \qquad \frac{1}{\sqrt{5}} (-2,1,0) = \left( \frac{-2}{\sqrt{5}}, \frac{1}{\sqrt{5}}, 0 \right) . \nonumber \]
Footnotes
[1] When Euclid defined angles in his Elements , the only angle he ever really defined was the right angle.


14.8: Putting It Together- Integrated Marketing Communications and Promotions
The emergence of digital technology, social media and mobile devices has led to significant changes in how consumers shop for and engage with brands—the emergence of an omni-channel, i.e. the varied interactions in the shopping process that consumers have pre-, during and post-purchase. The resulting proliferation of channels and fragmentation of target audiences means that advertising & promotional activity must be integrated together. The benefit is that this creates a multiplier effect, potentially maximizing impact on the target audience. But, it necessitates that the marketers selects the channels that are the best fit for their budget, strategy and the competitive situation. Further, they must ensure that all messaging is consistent, complementary and customized for the appropriate channel.


8.16F: Newly Discovered Eukaryotes
- Recognize Eukaryotic diversity
According to the Global Taxonomy Initiative and the European Distributed Institute of Taxonomy, the total number of species for some phyla may be much higher than what was known in 2010:10–30 million insects; (of some 0.9 million we know today) 5–10 million bacteria; 1.5 million fungi; of some 0.075 million we know today. The number of microbial species is not reliably known, but the Global Ocean Sampling Expedition dramatically increased the estimates of genetic diversity by identifying an enormous number of new genes from near-surface plankton samples at various marine locations.
From these studies, it is apparent that less than 1% of all species that have been described have been studied beyond simply noting their existence. The vast majority of Earth’s species are microbial. Contemporary biodiversity is “firmly fixated on the visible world”. For example, microbial life is metabolically and environmentally more diverse than multicellular life (e.g., extremophile). On the tree of life, based on analyses of small-subunit ribosomal RNA, visible life consists of barely noticeable twigs. The inverse relationship of size and population recurs higher on the evolutionary ladder.
Due to the advent of mass sequencing tools, thousands of new viral species have been identified in metagenomics studies, while at the same time hundreds of new viral species have been found. On top of that, there have been numerous fungal species identified. However, as suggested above most of the attention is given to large species, which represent a very small portion of the new species identified Even with this in mind, since the beginning of this century, 5 marsupial species, 25 primate, 1 elephant, 1 sloth, 3 rabbit, several rodent species, at least 30 new bat species have been discovered. On top of that several subspecies have been found. Considering how large an elephant is, this should point out how little we know about the numbers of microscopic eukaryotes that are yet to be discovered.
Of course we may never truly identify many eukaryotic species, since the rate of extinction has increased. Many extant species may become extinct before they are described.
LICENSES AND ATTRIBUTIONS
CC LICENSED CONTENT, SHARED PREVIOUSLY
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
CC LICENSED CONTENT, SPECIFIC ATTRIBUTION
- Cristae. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Cristae . License : CC BY-SA: Attribution-ShareAlike
- Crown group. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Crown_group . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote%23Phylogeny . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote%23Origin_of_eukaryotes . License : CC BY-SA: Attribution-ShareAlike
- cristae. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/cristae . License : CC BY-SA: Attribution-ShareAlike
- crown group. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/crown%20group . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote . License : Public Domain: No Known Copyright
- Order (biology). Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Order_(biology) . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote%23Origin_of_eukaryotes . License : CC BY-SA: Attribution-ShareAlike
- Domain (biology). Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Domain_(biology) . License : CC BY-SA: Attribution-ShareAlike
- kingdom. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/kingdom . License : CC BY-SA: Attribution-ShareAlike
- clade. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/clade . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote . License : Public Domain: No Known Copyright
- Biological classification L Pengo vflip. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Bi...engo_vflip.svg . License : Public Domain: No Known Copyright
- Opisthokonta. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Opisthokonta . License : CC BY-SA: Attribution-ShareAlike
- Flagellate. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Flagellate . License : CC BY-SA: Attribution-ShareAlike
- Opisthokonta. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Opisthokonta . License : CC BY-SA: Attribution-ShareAlike
- Opisthokonta. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Opisthokonta . License : CC BY-SA: Attribution-ShareAlike
- Opisthokonta. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Opisthokonta . License : CC BY-SA: Attribution-ShareAlike
- Opisthokonta. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Opisthokonta . License : CC BY-SA: Attribution-ShareAlike
- monophyletic. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/monophyletic . License : CC BY-SA: Attribution-ShareAlike
- flagellum. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/flagellum . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote . License : Public Domain: No Known Copyright
- Biological classification L Pengo vflip. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Biological_classification_L_Pengo_vflip.svg . License : Public Domain: No Known Copyright
- Flagellum (PSF). Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...llum_(PSF).png . License : Public Domain: No Known Copyright
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44593/latest/?collection=col11448/latest . License : CC BY: Attribution
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//biology/definition/genome-fusion . License : CC BY-SA: Attribution-ShareAlike
- endosymbiosis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/endosymbiosis . License : CC BY-SA: Attribution-ShareAlike
- symbiotic. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/symbiotic . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote . License : Public Domain: No Known Copyright
- Biological classification L Pengo vflip. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Biological_classification_L_Pengo_vflip.svg . License : Public Domain: No Known Copyright
- Flagellum (PSF). Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Flagellum_(PSF).png . License : Public Domain: No Known Copyright
- OpenStax College, Perspectives on the Phylogenetic Tree. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44593/latest...e_20_03_04.jpg . License : CC BY: Attribution
- OpenStax College, Perspectives on the Phylogenetic Tree. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44593/latest...e_20_03_03.jpg . License : CC BY: Attribution
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44616/latest...ol11448/latest . License : CC BY: Attribution
- OpenStax College, Biology. October 23, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44616/latest...ol11448/latest . License : CC BY: Attribution
- OpenStax College, Biology. November 14, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44629/latest...ol11448/latest . License : CC BY: Attribution
- phagocytosis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/phagocytosis . License : CC BY-SA: Attribution-ShareAlike
- multinucleate. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/multinucleate . License : CC BY-SA: Attribution-ShareAlike
- phagosome. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/phagosome . License : CC BY-SA: Attribution-ShareAlike
- amorphous. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/amorphous . License : CC BY-SA: Attribution-ShareAlike
- pellicle. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/pellicle . License : CC BY-SA: Attribution-ShareAlike
- taxis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/taxis . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote . License : Public Domain: No Known Copyright
- Biological classification L Pengo vflip. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Biological_classification_L_Pengo_vflip.svg . License : Public Domain: No Known Copyright
- Flagellum (PSF). Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Flagellum_(PSF).png . License : Public Domain: No Known Copyright
- OpenStax College, Perspectives on the Phylogenetic Tree. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44593/latest...e_20_03_04.jpg . License : CC BY: Attribution
- OpenStax College, Perspectives on the Phylogenetic Tree. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44593/latest...e_20_03_03.jpg . License : CC BY: Attribution
- OpenStax College, Characteristics of Protists. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44616/latest..._B23_02_02.jpg . License : CC BY: Attribution
- OpenStax College, Characteristics of Protists. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44616/latest..._B23_02_01.jpg . License : CC BY: Attribution
- Biodiversity. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Biodiversity . License : CC BY-SA: Attribution-ShareAlike
- Mammals described in the 2000s. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Mammals...d_in_the_2000s . License : CC BY-SA: Attribution-ShareAlike
- Biodiversity. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Biodiversity . License : CC BY-SA: Attribution-ShareAlike
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//biology/de...n/metagenomics . License : CC BY-SA: Attribution-ShareAlike
- extant. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/extant . License : CC BY-SA: Attribution-ShareAlike
- Eukaryote. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Eukaryote . License : Public Domain: No Known Copyright
- Biological classification L Pengo vflip. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Biological_classification_L_Pengo_vflip.svg . License : Public Domain: No Known Copyright
- Flagellum (PSF). Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Flagellum_(PSF).png . License : Public Domain: No Known Copyright
- OpenStax College, Perspectives on the Phylogenetic Tree. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44593/latest/Figure_20_03_04.jpg . License : CC BY: Attribution
- OpenStax College, Perspectives on the Phylogenetic Tree. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44593/latest/Figure_20_03_03.jpg . License : CC BY: Attribution
- OpenStax College, Characteristics of Protists. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44616/latest/Figure_B23_02_02.jpg . License : CC BY: Attribution
- OpenStax College, Characteristics of Protists. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44616/latest..._B23_02_01.jpg . License : CC BY: Attribution
- Undiscovered species chart. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...cies_chart.png . License : Public Domain: No Known Copyright
Key Points
- We know only a fraction of the number of species out there, and most of the attention has been given to large macroscopic species, a true understanding of microscopic eukaryotic life is not known.
- Even though mammals make up a small percentage of the species that are eukaryotes, several new mammalian species have been identified in the last decade.
- As extinction is increasing, we may never know all the eukaryotes that share the world with us.
Key Terms
- extant : Still alive; not extinct.
- metagenomics : The study of genomes recovered from environmental samples; especially the differentiation of genomes from multiple organisms or individuals, either in a symbiotic relationship, or at a crime scene.


10.3: Properties of Liquids
-
- Last updated
- Save as PDF
Learning Objectives
By the end of this section, you will be able to:
- Distinguish between adhesive and cohesive forces
- Define viscosity, surface tension, and capillary rise
- Describe the roles of intermolecular attractive forces in each of these properties/phenomena
When you pour a glass of water, or fill a car with gasoline, you observe that water and gasoline flow freely. But when you pour syrup on pancakes or add oil to a car engine, you note that syrup and motor oil do not flow as readily. The viscosity of a liquid is a measure of its resistance to flow. Water, gasoline, and other liquids that flow freely have a low viscosity. Honey, syrup, motor oil, and other liquids that do not flow freely, like those shown in Figure \(\PageIndex{1}\), have higher viscosities. We can measure viscosity by measuring the rate at which a metal ball falls through a liquid (the ball falls more slowly through a more viscous liquid) or by measuring the rate at which a liquid flows through a narrow tube (more viscous liquids flow more slowly).
The IMFs between the molecules of a liquid, the size and shape of the molecules, and the temperature determine how easily a liquid flows. As Table \(\PageIndex{1}\) shows, the more structurally complex are the molecules in a liquid and the stronger the IMFs between them, the more difficult it is for them to move past each other and the greater is the viscosity of the liquid. As the temperature increases, the molecules move more rapidly and their kinetic energies are better able to overcome the forces that hold them together; thus, the viscosity of the liquid decreases.
| Substance | Formula | Viscosity (mPa·s) |
|---|---|---|
| water | H 2 O | 0.890 |
| mercury | Hg | 1.526 |
| ethanol | C 2 H 5 OH | 1.074 |
| octane | C 8 H 18 | 0.508 |
| ethylene glycol | CH 2 (OH)CH 2 (OH) | 16.1 |
| honey | variable | ~2,000–10,000 |
| motor oil | variable | ~50–500 |
The various IMFs between identical molecules of a substance are examples of cohesive forces . The molecules within a liquid are surrounded by other molecules and are attracted equally in all directions by the cohesive forces within the liquid. However, the molecules on the surface of a liquid are attracted only by about one-half as many molecules. Because of the unbalanced molecular attractions on the surface molecules, liquids contract to form a shape that minimizes the number of molecules on the surface—that is, the shape with the minimum surface area. A small drop of liquid tends to assume a spherical shape, as shown in Figure \(\PageIndex{2}\), because in a sphere, the ratio of surface area to volume is at a minimum. Larger drops are more greatly affected by gravity, air resistance, surface interactions, and so on, and as a result, are less spherical.
Surface tension is defined as the energy required to increase the surface area of a liquid, or the force required to increase the length of a liquid surface by a given amount. This property results from the cohesive forces between molecules at the surface of a liquid, and it causes the surface of a liquid to behave like a stretched rubber membrane. Surface tensions of several liquids are presented in Table \(\PageIndex{2}\). Among common liquids, water exhibits a distinctly high surface tension due to strong hydrogen bonding between its molecules. As a result of this high surface tension, the surface of water represents a relatively “tough skin” that can withstand considerable force without breaking. A steel needle carefully placed on water will float. Some insects, like the one shown in Figure \(\PageIndex{3}\), even though they are denser than water, move on its surface because they are supported by the surface tension.
| Substance | Formula | Surface Tension (mN/m) |
|---|---|---|
| water | H 2 O | 71.99 |
| mercury | Hg | 458.48 |
| ethanol | C 2 H 5 OH | 21.97 |
| octane | C 8 H 18 | 21.14 |
| ethylene glycol | CH 2 (OH)CH 2 (OH) | 47.99 |
Surface tension is affected by a variety of variables, including the introduction of additional substances on the surface. In the late 1800s, Agnes Pockels, who was initially blocked from pursuing a scientific career but studied on her own, began investigating the impact and characteristics of soapy and greasy films in water. Using homemade materials, she developed an instrument known as a trough for measuring surface contaminants and their effects. With the support of renowned scientist Lord Rayleigh, her 1891 paper showed that surface contamination significantly reduces surface tension, and also that changing the characteristics of the surface (compressing or expanding it) also affects surface tension. Decades later, Irving Langmuir and Katharine Blodgett built on Pockels' work in their own trough and important advances in surface chemistry. Langmuir pioneered methods for producing single-molecule layers of film; Blodgett applied these to the development of non-reflective glass (critical for film-making and other applications), and also studied methods related to cleaning surfaces, which are important in semiconductor fabrication.
The IMFs of attraction between two different molecules are called adhesive forces . Consider what happens when water comes into contact with some surface. If the adhesive forces between water molecules and the molecules of the surface are weak compared to the cohesive forces between the water molecules, the water does not “wet” the surface. For example, water does not wet waxed surfaces or many plastics such as polyethylene. Water forms drops on these surfaces because the cohesive forces within the drops are greater than the adhesive forces between the water and the plastic. Water spreads out on glass because the adhesive force between water and glass is greater than the cohesive forces within the water. When water is confined in a glass tube, its meniscus (surface) has a concave shape because the water wets the glass and creeps up the side of the tube. On the other hand, the cohesive forces between mercury atoms are much greater than the adhesive forces between mercury and glass. Mercury therefore does not wet glass, and it forms a convex meniscus when confined in a tube because the cohesive forces within the mercury tend to draw it into a drop (Figure \(\PageIndex{4}\)).
If you place one end of a paper towel in spilled wine, as shown in Figure \(\PageIndex{5}\), the liquid wicks up the paper towel. A similar process occurs in a cloth towel when you use it to dry off after a shower. These are examples of capillary action —when a liquid flows within a porous material due to the attraction of the liquid molecules to the surface of the material and to other liquid molecules. The adhesive forces between the liquid and the porous material, combined with the cohesive forces within the liquid, may be strong enough to move the liquid upward against gravity.
Towels soak up liquids like water because the fibers of a towel are made of molecules that are attracted to water molecules. Most cloth towels are made of cotton, and paper towels are generally made from paper pulp. Both consist of long molecules of cellulose that contain many −OH groups. Water molecules are attracted to these −OH groups and form hydrogen bonds with them, which draws the H 2 O molecules up the cellulose molecules. The water molecules are also attracted to each other, so large amounts of water are drawn up the cellulose fibers.
Capillary action can also occur when one end of a small diameter tube is immersed in a liquid, as illustrated in Figure \(\PageIndex{6}\). If the liquid molecules are strongly attracted to the tube molecules, the liquid creeps up the inside of the tube until the weight of the liquid and the adhesive forces are in balance. The smaller the diameter of the tube is, the higher the liquid climbs. It is partly by capillary action occurring in plant cells called xylem that water and dissolved nutrients are brought from the soil up through the roots and into a plant. Capillary action is the basis for thin layer chromatography, a laboratory technique commonly used to separate small quantities of mixtures. You depend on a constant supply of tears to keep your eyes lubricated and on capillary action to pump tear fluid away.
The height to which a liquid will rise in a capillary tube is determined by several factors as shown in the following equation:
\[h=\dfrac{2 T \cos \theta}{r \rho g} \label{capillary} \]
In this equation, \(h\) is the height of the liquid inside the capillary tube relative to the surface of the liquid outside the tube, \(T\) is the surface tension of the liquid, \(θ\) is the contact angle between the liquid and the tube, \(r\) is the radius of the tube, \(ρ\) is the density of the liquid, and \(g\) is the acceleration due to gravity, 9.8 m/s 2 . When the tube is made of a material to which the liquid molecules are strongly attracted, they will spread out completely on the surface, which corresponds to a contact angle of 0°. This is the situation for water rising in a glass tube.
Example \(\PageIndex{1}\): Capillary Rise
At 25 °C, how high will water rise in a glass capillary tube with an inner diameter of 0.25 mm?
For water, T = 71.99 mN/m and ρ = 1.0 g/cm 3 .
Solution
The liquid will rise to a height h given by Equation \ref{capillary}:
\[h=\dfrac{2 T \cos \theta}{r \rho g} \nonumber \]
The Newton is defined as a kg m/s 2 , and so the provided surface tension is equivalent to 0.07199 kg/s 2 . The provided density must be converted into units that will cancel appropriately: ρ = 1000 kg/m 3 . The diameter of the tube in meters is 0.00025 m, so the radius is 0.000125 m. For a glass tube immersed in water, the contact angle is θ = 0°, so cos θ = 1. Finally, acceleration due to gravity on the earth is g = 9.8 m/s 2 . Substituting these values into the equation, and cancelling units, we have:
\[h=\frac{2\left(0.07199 kg / s^2\right)}{(0.000125 m )\left(1000 kg / m^3\right)\left(9.8 m / s^2\right)}=0.12 m =12 cm \nonumber \]
Exercise \(\PageIndex{1}\)
Water rises in a glass capillary tube to a height of 8.4 cm. What is the diameter of the capillary tube?
- Answer
-
diameter = 0.36 mm
Chemistry in Everyday Life: Biomedical Applications of Capillary Action
Many medical tests require drawing a small amount of blood, for example to determine the amount of glucose in someone with diabetes or the hematocrit level in an athlete. This procedure can be easily done because of capillary action, the ability of a liquid to flow up a small tube against gravity, as shown in Figure \(\PageIndex{7}\). When your finger is pricked, a drop of blood forms and holds together due to surface tension—the unbalanced intermolecular attractions at the surface of the drop. Then, when the open end of a narrow-diameter glass tube touches the drop of blood, the adhesive forces between the molecules in the blood and those at the glass surface draw the blood up the tube. How far the blood goes up the tube depends on the diameter of the tube (and the type of fluid). A small tube has a relatively large surface area for a given volume of blood, which results in larger (relative) attractive forces, allowing the blood to be drawn farther up the tube. The liquid itself is held together by its own cohesive forces. When the weight of the liquid in the tube generates a downward force equal to the upward force associated with capillary action, the liquid stops rising.


Page 5.3: The Human Population
Concepts of animal population dynamics can be applied to human population growth. Humans are not unique in their ability to alter their environment. For example, beaver dams alter the stream environment where they are built. Humans, however, have the ability to alter their environment to increase its carrying capacity, sometimes to the detriment of other species. Earth’s human population and their use of resources are growing rapidly, to the extent that some worry about the ability of Earth’s environment to sustain its human population. Long-term exponential growth carries with it the potential risks of famine, disease, and large-scale death, as well as social consequences of crowding such as increased crime.
Figure \(\PageIndex{1}\). Human population growth since 1000 AD is exponential.Human technology and particularly our harnessing of the energy contained in fossil fuels have caused unprecedented changes to Earth’s environment, altering ecosystems to the point where some may be in danger of collapse. Changes on a global scale including depletion of the ozone layer, desertification and topsoil loss, and global climate change are caused by human activities.
Human Population Growth
The fundamental cause of the acceleration of growth rate for humans in the past 200 years has been the reduced death rate due to changes in public health and sanitation. Clean drinking water and proper disposal sewage has drastically improved health in developed nations. Also, medical innovations such as the use of antibiotics and vaccines have decreased the ability of infectious disease to limit human population growth. In the past, diseases such as the bubonic plaque of the fourteenth century killed between 30 and 60 percent of Europe’s population and reduced the overall world population by as many as one hundred million people. Naturally, infectious disease continues to have an impact on human population growth, especially in poorer nations. For example, life expectancy in sub-Saharan Africa, which was increasing from 1950 to 1990, began to decline after 1985 largely as a result of HIV/AIDS mortality. The reduction in life expectancy caused by HIV/AIDS was estimated to be 7 years for 2005.
Figure \(\PageIndex{2}\). The time between the addition of each billion human beings to Earth decreases over time. (credit: modification of work by Ryan T. Cragun)Technological advances of the industrial age have also supported population growth through urbanization and advances in agriculture. These advances in technology were possible, in part, due to the exploitation of fossil fuels.
Age Structure, Population Growth, and Economic Development
The age structure of a population is an important factor in population dynamics. Age structure is the proportion of a population in different age classes. Models that incorporate age structure allow better prediction of population growth, plus the ability to associate this growth with the level of economic development in a region. Countries with rapid growth have a pyramidal shape in their age structure diagrams, showing a preponderance of younger individuals, many of whom are of reproductive age (Figure \(\PageIndex{3}\)).
Figure \(\PageIndex{3}\). Typical age structure diagrams are shown. The rapid growth diagram narrows to a point, indicating that the number of individuals decreases rapidly with age. In the slow growth model, the number of individuals decreases steadily with age. Stable population diagrams are rounded on the top, showing that the number of individuals per age group decreases gradually, and then increases for the older part of the population.This pattern is most often observed in underdeveloped countries where individuals do not live to old age because of less-than-optimal living conditions, and there is a high birth rate. Age structures of areas with slow growth, including developed countries such as the United States, still have a pyramidal structure, but with many fewer young and reproductive-aged individuals and a greater proportion of older individuals. Other developed countries, such as Italy, have zero population growth. The age structure of these populations is more conical, with an even greater percentage of middle-aged and older individuals. The actual growth rates in different countries are shown in Figure \(\PageIndex{4}\), with the highest rates tending to be in the less economically developed countries of Africa and Asia.
Figure \(\PageIndex{4}\). The percent growth rate of population in different countries is shown. Notice that the highest growth is occurring in less economically developed countries in Africa and Asia.Long-Term Consequences of Exponential Human Population Growth
Many dire predictions have been made about the world’s population leading to a major crisis called the “population explosion.” In the 1968 book The Population Bomb , biologist Paul R. Ehrlich wrote,
“The battle to feed all of humanity is over. In the 1970s hundreds of millions of people will starve to death in spite of any crash programs embarked upon now. At this late date nothing can prevent a substantial increase in the world death rate.”
While these predictions obviously didn’t bear fruit, the laws of exponential population growth are still in effect, and unchecked human population growth cannot continue indefinitely. Efforts to moderate population control led to the one-child policy in China, which imposes fines on urban couples who have more than one child. Due to the fact that some couples wish to have a male heir, many Chinese couples continue to have more than one child. The effectiveness of the policy in limiting overall population growth is controversial, as is the policy itself. Moreover, there are stories of female infanticide having occurred in some of the more rural areas of the country. Family planning education programs in other countries have had highly positive effects on limiting population growth rates and increasing standards of living.
In spite of population control policies, the human population continues to grow. The United Nations estimates the future world population size to be 11.2 billion people by the year 2100. There is no way to know whether human population growth will moderate to the point where the crisis described by Dr. Ehrlich will be averted. Another consequence of population growth is the change and degradation of the natural environment. Many countries have attempted to reduce the human impact on climate change by limiting their emission of greenhouse gases. However, a global climate change treaty remains elusive, and many underdeveloped countries trying to improve their economic condition may be less likely to agree with such provisions without compensation if it means slowing their economic development. Furthermore, the role of human activity in causing climate change has become a hotly debated socio-political issue in some developed countries, including the United States, despite the overwhelming scientific evidence. Thus, we enter the future with considerable uncertainty about our ability to curb human population growth and protect our environment to maintain the carrying capacity for the human species.
Contributors and Attributions
- Human Population Growth by OpenStax is licensed under CC BY 4.0. Modified from the original by Matthew R. Fisher.


3.5: Ukiyo-e (1636 )
- 120744
Introduction
In 1615, the Tokugawa clan consolidated their power in Japan. It moved the central government to Edo (modern Tokyo), the Edo period (also known as the Tokugawa period), and relative peace for the next 250 years. In contrast to the previous Momoyama period, the Tokugawa shogunate maintained stability, allowing for sustained prosperity based on an agrarian economy.
Prosperity and stability led to the growth of cities and sophisticated urban culture with a thirst for diversions from the mundane.
Theaters, tea houses, bars, and brothels were diversions for the popular cultures of the era. For taxation and control, the shogunate confined these activities within a few blocks in the city to an area known as the pleasure quarters. Cities such as Edo, Kyoto, and Osaka developed pleasure quarters, the centers of popular entertainment. The participants of Edo popular cultures were chōnin (townspeople), comprised primarily of merchants and artisans. [1]
In the hierarchy of Tokugawa society, merchants occupied the lowest rung, while samurais occupied the highest. Ironically, the merchant class gained the most financial power. Sumptuary laws and rigid stratification meant chōnin were unable to improve their social standing despite their education, skills, and wealth. As a result, they spent their leisure time and disposable income in literary and artistic pursuits while indulging in the pleasure quarter: theater, women, food, drinks, dancing, music, and sightseeing. These activities led to the rise of arts and cultures of the commoners (the middle class) as an alternative to the arts that high-ranking samurais patronized. The pictorial expression reflected this world of popular entertainment in these urban centers known as ukiyo- e .
“ Ukiyo ” (浮世) means the “floating world” while “e” (絵) means “picture(s).” The idea of the “floating world” is related to an older Buddhist concept of melancholia from the impermanence of human life on earth. Both concepts highlighted the fleeting, untethered existence where nothing lasted – fashion, beauty, seasons, a night of passion. The entertainment world of the pleasure quarter thrived on novelty and ever-changing trends. In the Edo period, the term “floating” emphasized savoring the present moment in a carefree realm, forgetting the mundane constraints of everyday life. Pictures of the floating world – ukiyo-e -- were visual records of the activities of the chōnin and the people who worked to entertain them.
Hishikawa Moronobu
The series
The Appearance of Yoshiwara
(3.5.1)
by Hishikawa Moronobu (1618-1694) shows a brothel interior. (Yoshiwara was the pleasure quarter of Edo). The patrons sat down towards the right, just in front of a tree. They enjoyed a dance by a courtesan while musicians accompanied her with samisens and a drum; attendants on the side waited, and served food and drinks. In addition to genre scenes of the pleasure quarters, subject matters of ukiyo-e include beautiful women (often courtesans), kabuki actors, scenes in kabuki plays, erotica, ghost stories, and folk tales. In the 19
th
century, as domestic travel and tourism increased, landscapes (images of beautiful places) also became popular.
Although the term “ukiyo-e” today is synonymous with Japanese woodblock prints from the seventeenth to the nineteenth century, representations of the floating world first appeared in paintings in the early seventeenth century. Since paintings could not keep up with the insatiable demands for such images, woodblock prints became the primary medium for ukiyo-e starting in the second half of the seventeenth century. With woodblock printing, multiple editions of the same image were inexpensively reproduced to satisfy the masses. Japanese woodblock printing was essentially a collaborative process between a publisher, an artist/designer, a block carver, and a printer. The artists drew or painted an image in consultation with the publisher, who determined whether mass production of the image would be profitable. The publisher also gathered the funds and set up all personnel involved in the production of the prints. The block carver transferred the artist’s drawing onto a block of wood (typically cherry wood), and the printer applied the dyes to the printing surface and created the print. Unfortunately, the names of the last two professionals were not recorded for posterity; the print only stated the name of the publisher and the artist. The names discussed here are the names of the artists/designer of the prints. [2]
Suzuki Harunobu
Initially, prints were only black and white, as seen in Hishikawa Moronobu’s print above. As printing techniques evolved, polychrome prints appeared in the middle of the eighteenth century. The influence of chōnin in this development cannot be understated; Suzuki Harunobu (1724-1770) created the first multicolored calendar in 1764 under the private commission of a group of wealthy upper-class chōnin and samurai. With the financial support from his patrons, he experimented with color dyes, woods, and printing techniques to achieve a print with ten colors. Such a print was unheard of at the time, but it quickly became the inspiration for other artists and publishers.
Harunobu mainly represented beautiful women or women with their paramours. Both males and females have slight figures; the outdoor settings are sparse yet idyllic . Lovers Walking in the Snow (3.5.2) portrays a romantic scene of two people sharing a large umbrella as they walked in the snow. The contrast between the man’s black attire and the pale snow centers the composition. The man turns solicitously towards his companion as she demurely looks down and leans her head towards him. This subtlety and lyricism make Harunobu’s prints so appealing.
 Harunobu’s detailed rendering of furniture, tea utensils, room partition, and windows gives his viewers s a glimpse of the interior of a Japanese house in this period.
Torii Kiyonaga
Images of beautiful women is a category in ukiyo-e knowns as bijinga . The subject was mainly courtesans of the pleasure quarters, although some artists also represented local beauties outside the red-light district. In some ways, Harunobu popularized this category by rendering lithe frames in colorful outfits and tranquil backgrounds. Another artist considered the next master in this category is Torii Kiyonaga (1752-1815). In contrast to Harunobu’s slender figures, Kiyonaga’s women are tall and statuesque; they fill their space with grace and confidence. Kiyonaga also dealt with complex compositions by having more than two characters in a diptych or a triptych.
Enjoying the Evening Cool on the Banks of the Sumida River (3.5.4) is a diptych; however, each image can exist independently without the other. The left image depicts three women enjoying the breeze by the river while sitting and standing by a bench. The seated lady in the darkest outfit is the most captivating as she takes in her surroundings -- the breeze and the view -- with joyful abandon.  Following her gaze and posture, viewers can see a row of houses across the river, adding depth and an overall riverine atmosphere to the image.
Kitagawa Utamaro
Kitagawa Utamaro (1753-1806) added another level of excitement to the bijinga category by depicting beautiful women close-ups, showing their heads and torsos only. Hanamurasaki of the Tamaya , (3.5.5) from the series Seven Komachi of the Pleasure Quarters is an example of this format. Utamaro captured the celebrated courtesan Hanamurasaki poised with her brush, wetting it on her lips just before she was about to write her letters. The close-up format gives the artist ample opportunity to lavish details on the elaborate coiffure of the courtesan and the exquisite patterns of her kimono.
Utamaro’s primary interests were not in patterns and decorations; he was interested in depicting the many daily activities of Edo period women. For example, an intimate portrayal of motherhood can be seen in Midnight: Mother and Sleepy Child (3.5.6) from the series Customs of Beauties Around the Clock . Other activities portrayed include women doing their hair and make-up, reading books and letters, observing insects, or watching themselves in a mirror. Utamaro’s keen eyes and genuine interest made him the foremost artist in representing Edo’s beauties.
Kabuki was -- and still is -- an exciting performance; actors’ gestures and expressions are exaggerated, and stages have trapdoors and rotating mechanisms to facilitate a dramatic entry and swift character transformation. Interior of a Kabuki theater (3.5.8) demonstrates the details of the theater.
Kabuki theater and the woodblock printing industry were interdependent. The theater industry relied on inexpensive printed brochures, playbills, and posters to promote their plays. At the same time, Kabuki actors provided the subject matter for yakusha-e (actor prints), a source of creativity and income for the ukiyo-e industry. Actor prints were similar to posters of movie stars today; adoring fans purchased and collected them. Efficiency in printing and competition among publishers kept the price low enough for everybody to own a print at the cost of a bowl of noodles. Ichikawa Danjūrō V as Hannya no Gorō in Shibaraku (3.5.7) was an advertisement for the theater.
Katsukawa Shunsho
Katsukawa Shunsho (1726 -1793) was known as a prominent artist from a well-known school, and he studied with two famous ukiyo-e artists. Shunsho was celebrated for his depictions of Kabuki theatre actors. Kabuki was -- and still is -- an exciting performance; actors' gestures and expressions are exaggerated, and stages have trapdoors and rotating mechanisms to facilitate a dramatic entry and swift character transformation. Interior of a Kabuki theater (x.x) demonstrates the details of the theater. Kabuki theater and the woodblock printing industry were interdependent. The theater industry relied on inexpensive printed brochures, playbills, and posters to promote their plays. At the same time, Kabuki actors provided the subject matter for yakusha-e (actor prints), a source of creativity and income for the ukiyo-e industry. Actor prints were similar to posters of movie stars today; adoring fans purchased and collected them. Efficiency in printing and competition among publishers kept the price low enough for everybody to own a print at the cost of a bowl of noodles. Ichikawa Danjūrō V as Hannya no Gorō in Shibaraku (x.x) was an advertisement for the theater.
In yakusha-e , an actor appears in his role. The actor Ichikawa Danjūrō V (x.x) was captured in his role as Hannya no Gorō in the play Shibaraku ; for this role, he wore an elaborate wig, and his face was painted in red and white. He cut a menacing figure with one arm, lifting his sword and sporting a dramatic frown. Shunsho was known for creating more realistic portrayals of actors and close-up portraits consisting of the heads and torsos of actors.
Toshūsai Sharaku
Toshūsai Sharaku (active 1794–1795) designed the most expressive depictions of Kabuki actors in the head-and-torso-only format. With realism unusual for the time, Sharaku managed to capture the personality traits of Kabuki characters. The print of Ōtani Oniji III as Manservant Edobei (3.5.9) is iconic among Sharaku prints. Sharaku utilized hand gestures and facial expressions to indicate Edobei’s wicked dramatis persona. Edobei was about to rob someone; Sharaku accentuated this impeding action by showing him leaning forward with his two hands opening wide, ready to grab its prey.  Sharaku used only three lines to represent the pent-up tension of the moment. One curving line indicated his taut jaw, and two “S” shape lines on Edobei’s temple indicated two strands of hair escaped from his otherwise pristine coiffure. The lines are so fine that from afar, viewers may have mistaken them for trails of sweat. Capitalizing on this visual ambiguity, Sharaku managed to represent both signs of stress with an economy of lines.
Very little is known about Sharaku. Only around 140 prints by him were produced in the ten months between 1794-95. After that, the name and the stark realism a la Sharaku disappeared. Some experts believe that Sharaku’s prints were too unflattering, closer to caricatures than portraitures, and thus, they lost their customer base. As for the artist, some speculated that the name is an alias of another artist, and it ceased to be used once people lost interest in Sharaku-style prints.
In the early 19 th century, landscape prints became popular due to the rising interest in travel and tourism. People wanted to learn about other parts of Japan beyond their domain. Landscape prints functioned as souvenirs for travelers who had visited a site, or the prints reminded people of places to stay in the future. Two of the most significant landscape print artists were Katsushika Hokusai (1760-1849) and Utagawa Hiroshige (1797-1858). The prints of both artists are well known in the Western world, inspiring modern artists such as Monet, Whistler, Degas, and, Van Gogh in the second half of the nineteenth century and beyond.
Katsushika Hokusa
The challenge in creating prints in a series is creating variations around a theme, variations intriguing enough to keep customers buying from the same series. What makes 36 Views of Mount Fuji interesting is the variety of human activities set against different landscape views in each print. In Nakahara in Sagami Province (3.5.10), the main interest is people from different walks of life converging around a bridge over a small river, some locals and others travelers. The fishermen on the left, the woman with a baby and round tray on her head, and the peddler near her were local people, while others were travelers; generally, pilgrims as the area is near a religious pilgrimage site. Mount Fuji -- visible at the center – does not provide a stage for a more exciting composition as in The Great Wave . In this print, the joy stems from learning about the slice of life in everyday nineteenth-century Japan.
Hokusai’s The Great Wave of Kanagawa (3.5.11) is arguably the most iconic, well-known ukiyo-e print in the world. With limited colors and shapes, Hokusai created a masterpiece of composition. The print came from a series of 36 Views of Mount Fuji , and although the mountain is barely visible in the background, Hokusai ensured that the viewers would not miss it. The giant swirling wave is a half-circle that, if completed, frames the mountain. The stylized fringes of the waves form menacing claw-like talons that threatened the boats yet also pointed towards Mount Fuji. The larger triangle created by the wave in the foreground mimicked the small triangle of Mount Fuji. The primary color in the print is Prussian blue, a synthetic pigment recently introduced in Japan. Hokusai was one of the first artists who embraced this color; it appeared on almost every print in this series.
Utagawa Hiroshige
In the Edo period, Kyoto and Tokyo were connected by two roads. Utagawa Hiroshige (1797-1858) traveled through the Tōkaidō road in 1832, making notes and sketching sceneries along the way; these became the basis of 53 Stations of the Tōkaidō . This is arguably the only series that can match the virtuosity of Hokusai’s 36 Views of Mount Fuji . Both series tell the stories of human lives in landscape settings, and both demonstrate compositional mastery. Hiroshige’s series is more lyrical, representing different weathers and various times of the day, and changing seasons factors influenced the mood of the print. The frenzy in Sudden Shower at Shōno (3.5.12) is palpable; people are running up and down the slope to avoid the summer downpour. Multiple diagonals – some as shapes and others as lines – in different directions add to the frenetic energy of the moving bodies.
In contrast, calm and stillness pervade Evening Snow at Kanbara (3.5.13). Hiroshige prioritized mood over objective veracity in depicting this site as Kanbara is a coastal town in Shizuoka Prefecture, and it rarely snows there.
Utagawa Kuniyoshi
Utagawa Kuniyoshi (1798-1861) is considered to be one of the last ukiyo-e masters and portrayed not only bijin-ga and kabuki actors, but a wider set of samurai heroes, cats of all types, and mythical animals, frequently including satire in his images. He learned to draw at an early age and quickly became an apprentice of one of the masters in the Utagawa school. He studied European art and regularly arranged his images in landscape orientation.
Onmayagashi in Edo (3.5.14) depicts three men struggling to stay dry under one umbrella passing the fisherman casually walking in the rain who is trailed by a single man carrying three umbrellas, rain splashing everywhere. Kuniyoshi often added subtle humor to his works. He used muted colors of the shadowy background and boats, bringing the activity of the scene to the foreground.
A Lovely Garland (3.5.15) is based on the story, The Great Woven Cap , the adventures of an ancient courtier. A diver recovered jewels from the dragon king who lived in the sea and with a dagger in hand, battles the octopus. Kuniyoshi uses tones of blue for the background to set off the ongoing battle, contrasting the diver’s elegantly patterned robes.
Ukiyo-e prints were very popular, a small piece of art available to all classes of people, not restricted as expensive paintings. Thousands of prints were made; however, the survival rate is limited because of the materials used to make the prints. The dyes faded from exposure to light and the paper deteriorated from handling, poor storage, or contact with acidic materials. The term ukiyo-e translates into ‘pictures of the floating world’, and the surviving prints give an insight into the Edo world, environment, and lives of the period; the samurai warriors, the kabuki theatre and actors, the rise of the courtesans and the engaging landscape of Japan.
[1] Lower ranking samurais did frequent the pleasure quarters, associated with merchants and artisans, and participate in these activities. However, the term chōnin typically refers to non-samurai participants of urban popular cultures.
[2] Regarding artist names: Full name will be used for the first time. After that the artists will be referred to by their first names according to the typical convention of referring to Ukiyo-e artists.


5.12: Other Criminological Theories
-
- Last updated
- Save as PDF
- Alison S. Burke, David Carter, Brian Fedorek, Tiffany Morey, Lore Rutz-Burri, & Shanell Sanchez
- Southern Oregon University via OpenOregon
Social Reaction Theories
Initially a response to popular theories at the time, social reaction theories change the focus. Instead of focusing on the offender, social reaction theories concentrate on those people or institutions who label offenders, react to offenders, and want to control offenders. Grounded in symbolic interaction, social reaction theories emphasize how meanings are constructed. Words carry power and meaning. Thus, if someone is labeled as a criminal, it carries a connotation by other people. How will others view him or her with that label? How will society? People will act towards thing by the meanings attached. These meanings can be culturally created through interactions with peers. Perhaps it is why some of us like to wear certain name brand clothes or drive certain name brand vehicles. Those names have “meaning”; they can be cool, reliable, etc. They have status and meaning.
Not everyone who commits a crime is caught.  Why? Labeling theories sought to explain this phenomenon. In general, labeling theorists point to the social construction of crime, which varies over time and place. For instance, marijuana use is federally prohibited, but more and more states are legalizing recreational use. The same behavior can be legal in Oregon but illegal in New York.
Furthermore, labeling theorists emphasize the process of being labeled and treated as a criminal; it can have deleterious effects. If a person is bombarded with a particular label, he or she may adopt that label. If a person was told over and over again that he or she was funny, that person might try to be funnier. What if a person were told over and over again he or she would amount to nothing? Perhaps, he or she would internalize that label and become nothing since others expect that.
Braithwaite (1989) applied some of these ideas in his reintegrative shaming. Not all social reactions are ruthless; in fact, some of them are beneficial. [1] Hence, it is the quality of social responses that is significant. We shame individuals to show disapproval. According to Braithwaite, shaming can be reintegrative or stigmatizing. Reintegrative shaming centers on forgiveness, love, and respect. Ideally, we want to reintegrate the person back into the community by removing the label. However, in some societies, like the United States, stigmatizing shaming reigns supreme. Stigmatizing shaming uses formal punishment, which degrades a person’s bond to his or her community. It is counter-productive and tends to shun the offender. For example, in some states, convicted offenders are required to self-identify as a felon on job applications. Do you think this helps their cause to reintegrate successfully into society? Perhaps not. Even though they may have “served their time,” they are still labeled as a criminal and punished further. Stigmatizing shaming propels people towards crime whereas reintegrative shaming seeks to correct the behavior through respect and empathy. Critical Theories
Critical Theories
Critical theories originated during the 1960-1970s in the United States. Immediately following World War II, Americans expected the United States to thrive economically, culturally, and politically. Massive political turmoil inside and outside of the country created a generation of scholars who became “critical” of society and more “traditional” theories of crime. According to Cullen, Agnew, and Wilcox (2018), critical theories share some five central themes. First, to understand crime, one must appreciate the fusion between power and inequality. People with power, political and economic, have an enormous advantage in society. [2] Second, crime is a political concept.  The poor are injured the most by the enforcement of laws, while the affluent (i.e., powerful) are treated leniently. Third, the criminal justice system and its agents serve the ruling class, the capitalists. As Jeffrey Reiman’s (2004) book titled it, The Rich Get Richer and the Poor Get Prison. [3] Fourth, the root cause of crime is capitalism because capitalism ignores the poor and their atrocious living conditions. Capitalism demands profits and growth over values and ethically considerations. Perhaps this is why crimes of the streets are punished more severely than crimes of the suites. Finally, critical theories believe the solution to crime is a more equitable society, both politically and economically.
It is beyond the scope of this book to delve into all of the possible explanations of crime. Some theories try to explain the differences between male offending and female offending through feminist theories of crime. Other theories have integrated or combined concepts from other theories to create a newer model of explanation.
Gender and Crime Exercise
Write about how you were raised and how sex/gender roles were reinforced through school, family, culture, etc. Do you think men/boys/males are more criminal because of their biology or because of cultural expectations of men/boys/males versus women/girls/females? You should support your claims with personal, vicarious, or well-known examples.
- Braithwaite, J. (1989). Crime, shame and reintegration . New York, NY: Cambridge University Press. ↵
- Cullen, F.T., Agnew, R., & Wilcox, P. (Eds). (2018). Criminological theory: Past to present (6th ed.). New York, NY: Oxford University Press. ↵
- Reiman, J. (2004). The rich get richer and the poor get prision: Ideology, class, and criminal justice. New York, NY: Pearson. ↵


11.4: Positron Emission
A positron , also called an antielectron , is an exotic bit of matter, or more correctly, an example of antimatter . A positron is the antimatter equivalent of an electron. It has the mass of an electron, but it has a charge of +1. Positrons are formed when a proton sheds its positive charge and becomes a neutron, as shown below:
\[_{1}^{1}\rho \rightarrow +_{+1}^{0}\beta +_{0}^{1}n \nonumber \]
Again, in the nuclear equation for positron emission, the sum of protons (atomic numbers) on the right equals the number of protons on the left and the masses all equal one. When an element emits a positron, the identity of the element changes to the one having one fewer protons on the periodic table. An example of a nuclear equation showing positron emission is shown below:
\[_{6}^{11}C \rightarrow +_{+1}^{0}\beta +_{5}^{11}B \nonumber \]
Boron has one fewer protons in its nucleus than carbon, but the mass is unchanged because the proton has been replaced by a neutron.
\[_{9}^{18}F \rightarrow +_{+1}^{0}\beta +_{8}^{18}O \nonumber \]
Positron emission from Fluorine-18, as shown above , has become an important medical diagnostic tool; Positron Emission Tomography (a PET scan). The heart of this technique is based on the fact that positrons undergo instant annihilation when they collide with an electron (an example of matter-antimatter annihilation ). When this occurs, two high-energy gamma rays are produced and exit the scene of the annihilation in exactly opposite directions. During a PET scan, a patient is given an injection containing fluorodeoxyglucose (FDG), a sugar analog. The glucose analog is absorbed by metabolically active cells, where the FDG accumulates and undergoes positron decay. After a short waiting period, the patient is scanned using a circular array of gamma-radiation detectors. The fact that the gamma rays are emitted in opposite directions allows the attached computer to “draw a line” through the patient, where the line passes through the point of annihilation. Because this occurs through many directions, the exact location of the emission can be accurately calculated and then imaged as a three-dimensional picture showing the intensity of the emission.


1.8: Ethers
- How to name ethers.
In organic chemistry, any alkyl group can be abbreviated as R . An ether, in which the carbons of two alkyl groups are linked to the same oxygen, can be abbreviated as R-O-R.
This molecule is often referred to simply as “ether”. Its common name is diethyl ether, and its IUPAC name is ethoxyethane.
The common name of this molecule is ethyl methyl ether, and its IUPAC name is methoxyethane.
1. What are the common and IUPAC names of Molecule A?
2. The name of this molecule is 2-methoxy-3-methylbutane. Number the carbons.
3. Number the carbons. What is the name of Molecule B?
4. The name of this molecule is 4-butoxy-2-butanol. Number the carbons.
5. Number the carbons. What is the name of Molecule C?
6. What additions do we make to our existing naming rules to name ethers?
7. Write the steps that you use to name an ether in order, as instructions for a student who doesn't know how to do it.
8. Draw any ether and go through the steps in naming your molecule.


3: Middle Childhood
Middle childhood is the period of life that begins when children enter school and lasts until they reach adolescence. Think for a moment about children this age that you may know. What are their lives like? What kinds of concerns do they express and with what kinds of activities are their days filled? If it were possible, would you want to return to this period of life? Why or why not? Early childhood and adolescence seem to get much more attention than middle childhood. Perhaps this is because growth patterns slow at this time, the id becomes hidden during the latent stage, according to Freud, and children spend much more time in schools, with friends, and in structured activities. It may be easy for parents to lose track of their children’s development unless they stay directly involved in these worlds. I think it is important to stop and give full attention to middle childhood to stay in touch with these children and to take notice of the varied influences on their lives in a larger world.
-
- 3.1: Introduction to Middle Childhood
- Middle childhood is the period of life that begins when children enter school and lasts until they reach adolescence. Growth patterns slow at this time, the id becomes hidden during the latent stage, according to Freud, and children spend much more time in schools, with friends, and in structured activities. It may be easy for parents to lose track of their children’s development unless they stay directly involved in these worlds.
-
- 3.2: Physical Development
- Children tend to slim down and gain muscle strength and lung capacity during middle childhood, making it possible to engage in strenuous physical activity for long periods of time. The brain reaches its adult size at about age 7 so the school-aged child is better able to plan, coordinate activity using both left and right hemispheres of the brain, and to control emotional outbursts.
-
- 3.3: Cognitive Development
- From ages 7 to 11, the school-aged child is in what Piaget referred to as the concrete operational stage of cognitive development. The child can use logic to solve problems tied to their own direct experience but has trouble solving hypothetical problems or considering more abstract problems. The child uses inductive reasoning which means thinking that the world reflects one’s own personal experience.
-
- 3.4: Kohlberg's Stages of Moral Development
- Kohlberg called a self-centered approach to moral decision-making pre-conventional moral development. Conventional moral development referred to being able to determine right from wrong based on what other people think. Post-conventional moral development is based on a concern for others; for society as a whole or for an ethical standard rather than a legal standard.
-
- 3.5: Developmental Problems
- Children’s cognitive and social skills are evaluated as they enter and progress through school. Evaluation and diagnosis of a child can be the first step in helping to provide that child with the type of instruction and resources needed. It is important to consider that children can be misdiagnosed and that once a child has received a diagnostic label, the child, teachers, and family members may tend to interpret actions of the child through that label.
-
- 3.6: Learning and Intelligence - Schools and Testing
- Intelligence tests and psychological definitions of intelligence have been heavily criticized since the 1970s for being biased in favor of Anglo-American, middle-class respondents and for being inadequate tools for measuring non-academic types of intelligence or talent. Achievement tests are used to measure what a child has already learned. Aptitude tests are designed to measure a student’s ability to learn or to determine if a person has potential in a particular program.
-
- 3.7: Psychosocial Development
- Children in middle childhood have a more realistic sense of self than do those in early childhood. Contemporary children also receive messages from the media about how they should look and act. According to Erikson, children in middle childhood are very busy or industrious. They are constantly doing, planning, playing, getting together with friends, achieving. This is a very active time and a time when they are gaining a sense of how they measure up when compared with friends.


3.11: Assertive Communication
The skill of assertiveness is important to nurses. Nurses are expected to be the patients’ advocates. So, they need to have the assertive communication skills in order to be able to be patients’ advocates. Assertiveness enables a person to be honest with him/herself and in relationships with others. Assertiveness helps to enhance relationships, avoid power games and is a vehicle for clear outcomes. Hargis as cited by van Niekerk identifies four elements of assertive communication [ 8 ]:
- Content – where the rights of the people involved are embedded gently in the statement. This could be done by using an explanation, empathy for the listener, and praise for the listener, an apology for the consequence for the listener or a compromise that is favorable to both people.
- Covert elements – where the speaker is able to recognize their rights and the rights of the listener in the communication process. These include respect, expressing feelings, having your own priorities, being able to say ‘no’, being able to make mistakes and choosing to say nothing.
- Process – concerned with how people express themselves assertively. Is their body language, intonation and choice of language reflective of a confident assertive person? Are the processes that make up communication congruent, in keeping with what is being said? The process also involves managing the setting so that people are not embarrassed, or the noise levels are kept to a minimum. Increasing the likelihood of assertive communication happening again involves feedback to the listener to show that their accomplishment is appreciated.
- Non-verbal cues – gesture, touch, proxemics and posture – also need to reflect confidence, regard and respect for self and others.


8.11: Editing Files and Creating Functions (PC)
You should write a MATLAB program whenever you anticipate executing some sequence of statements several times or again in a later session. On an IBM PC, you may use any text editor to write a program, as long as the file can be saved in ASCII format without the control codes used by most word processors. Applicable text editors include Edix, Wordstar, XTree Pro, and Turbo Pascal's editor.
Editing Files
If you have enough memory, you can run your editor without leaving MATLAB by using the exclamation point (!), like this:
≫ !EDIX
The exclamation point may be used to execute any DOS command or program from MATLAB. When the command or program finishes, your MATLAB variables are just as you left them. Use your editor to write program lines just as you would type them in MATLAB's command mode. Then save the file with extension
.m
in the directory where you will run MATLAB. Such MATLAB programs are called m-files. You may run your m-file by typing the file name (without the
.m
extension) at the MATLAB command prompt (\(\gg\)).
Script Files
There are two kinds of m-files, called
script files
and
functions
. Running a script file is exactly like typing the commands it contains at the \(\gg\) prompt. Your m-file will automatically be a script file unless you specify otherwise, as described later. Practice by entering, saving, and running
plotsin.m
as listed next:
t = -6:.2:6; y = sin(t); plot(t,y) title('SINE') pause grid xlabel('t') ylabel('sin(t)')
When the pause is executed, you will need to press a key to go on. If you type whos after running
plotsin
, you will see that the variables
t
and
y
remain in memory. Comments are important to a script file. They are marked with the symbol %. Anything following this symbol on a line is assumed to be a comment and is ignored by the MATLAB program interpreter.
Functions
Functions differ from script files in that they have designated input and output variables. Any other variables used within a function are local variables, which do not remain after the function terminates and which have no effect on variables outside the function. Many of the functions supplied with MATLAB are actually m-files. A good example is
triu.m
:
≫ type triu
The word function at the beginning of the file makes it a function rather than a script file. The function name in this line must match the file name. The input variables of
triu
are
x
and
k
, meaning that the first input argument will be referred to as
x
and the second as
k
within the function. Likewise, the function line designates
y
as the output. There is nothing special about the variable names
x
,
k
, and
y
when the function is used. It is only that whatever inputs and output you use will be referred to as
x
,
k
, and
y
inside the function. The variables
m
,
n
,
j
, and
i
are created temporarily when
triu
runs and disappear when it terminates. They are local variables and have no effect on variables with the same names outside the function. In contrast, a script file has no local variables and does no substitution of input and output variable names.
As an exercise, enter and save the function
perp.m
:
function y=perp(x) % PERP(x) is a complex number perpendicular to x. j = sqrt(-1); y = j*real(x)-imag(x);
Evaluate
perp
on various complex numbers. Replace the last line by
y = x*j;
. Do you get the same result? Why?
Printing Files and Graphics
To display an m-file on the screen, use the instruction
≫ type filename
. To make a copy at a printer, use the DOS command
≫ !print filename.m
. Graphics hardcopy is available through the commands meta and
gpp
. See the MATLAB manual for more information.


7.2: Amino acids
- Define and distinguish amino acids, \(\alpha\)-amino acids, and proteinogenic amino acids.
- Draw Fisher projections and assign D/L or R/S stereodescriptors to proteinogenic amino acids.
- Understand the classification of proteinogenic amino acids based on the characteristics of the side chain.
- Define isoelectric point and understand the ionization states of amino acids under physiological conditions.
- Define essential amino acids and their sources.
What are amino acids?
Amino acids are the building blocks of proteins, i.e., they are the monomers of proteins. Amino acids are organic compounds that contain both an amine (\(\ce{-NH2}\)) and a carboxylic acid (\(\ce{-COOH}\)) group in the same molecule. For example, alanine, shown on the right, is an amino acid. Proteins contain a subclass of amino acids called
an amine (\(\ce{-NH2}\)) on the \(\alpha\ce{C}\) to \(\ce{-COOH}\) group, i.e., both the amine (\(\ce{-NH2}\)) group and the carboxylic acid (\(\ce{-COOH}\)) group are attached to the same \(\ce{C}\). For example, alanine is an
Proteinogenic amino acids are a subclass of \(\alpha\)-amino acids incorporated into proteins during biosynthesis. Twenty proteinogenic amino acids are usually present in proteins, and two additional are included in exceptional cases. This chapter's word "amino acid" refers to the 20 standard proteinogenic amino acids.
Configuration of \(\alpha\ce{C}\) of an
the \(\alpha\ce{C}\) of \(\alpha\)-amino acid have four different groups attached to it: amine (\(\ce{-NH2}\)), carboxylic acid ((\(\ce{-COOH}\)), hydrogen (\(\ce{-H}\)), and alkyl side chain (\(\ce{-R}\), as shown in Figure \(\PageIndex{1}\). There is one exception "glycine" that has two \(\ce{H's}\) at the \(\alpha\ce{C}\)
Three groups, i.e., \(\ce{-NH2}\), (\(\ce{-COOH}\), and \(\ce{-H}\) are present in all
, and the fourth group, i.e., the side chain (\(\ce{-R}\)) varies in differentThe \(\ce{-COOH}\) is acidic (pK a 2-5) and the \(\ce{-NH2}\) is basic (pK a ~10). The \(\ce{-COOH}\) loses its proton and exists as a carboxylate ion ( \(\ce{-COO^{-}}\); the \(\ce{-NH2}\) gains a proton and exists as ammonium ion ( \(\ce{-NH3^{+}}\) at physiological pH of 7.4. These groups ionize the same way if present in the side chain in addition to the \(\alpha\ce{C}\).
Compounds that have a positive charge on one atom and a negative charge on another atom in the same molecule are called zwitterions . Most of the
Figure \(\PageIndex{1}\).A \(\ce{C}\) with four different groups is a chiral center.
\(\alpha\ce{C}\) of \(\alpha\ce{C}\) "glycine" that has two \(\ce{H's}\) at the \(\alpha\ce{C}\). presented in Fisher projections with the \(\ce{C}\)-chain placed vertically and the \(\ce{-NH2}\) and \(\ce{-H}\) bonds shown horizontally, as shown in Figure \(\PageIndex{1}\).The
Proteinogenic amino acids are L-amino acids, with the exception of glycine, which is not chiral. In the R/S system, proteinogenic amino acids are ( S ) at the \(\alpha\ce{C}\), with the exception of "cysteine" being ( R ) and glycine not chiral.
The
Figure \(\PageIndex{2}\) roteinogenic amino acids are classified based on the nature of the side chain.Classification of proteinogenic amino acids
\(\alpha\)-Amino acids are classified based on the hydrophobicity of their side chain.
Hydrophobic means "water fearing", i.e., molecules or entities that tend to repel water, not dissolve, or not wetted by water. Hydrophilic means "water-loving", i.e., molecules or entities that tend to mix with, dissolve, or be wetted by water.
Hydrocarbons and other non-polar compounds are hydrophobic compounds that do not dissolve in water. Polar or ionic compounds are usually soluble in water. Based on these criteria, amino acids are classified into the following classes, shown in Figure 7.2.2.
- Nonpolar amino acids with an aliphatic side chain are hydrophobic. Methionine having a thioether group in the side chain is non-polar and placed in this group.
- Nonpolar amino acids with an aromatic side chain are hydrophobic. Tyrosine has phenol as part of its side chain, but it is hydrophobic and placed in this group.
- Polar neutral amino acids with a polar neutral side chain are hydrophilic. This class of side chain contains alcohol (\(\ce{-OH}\)) or amide \(\ce{-CONH2}\) groups in their side chain that hydrogen bond with water but do not ionize.
-
Amino acids in a special class include cysteine, glycine, and proline.
- Speciality of cysteine is that its thiol {\(ce{-SH}\) group is easily oxidized forming is a disulfide (\(\ce{S-S}\) bond, which is the only covalent bond in protein besides the amide bonds. Cystein is classified as nonpolar and hydrophobic because {\(ce{-SH}\) is a nonpolar group.
- Glycine is the only proteinogenic acid that has no chiral center. The \(\ce{-H}\) side chain places it at a borderline between hydrophilic and hydrophobic categories, it is considered neutral.
- Proline is the only amino acid with a secondary \(\alpha\)-amine group. The side chain is a five member ring with \(\ce{N}\) of \(\alpha\)-amine as part of the ring. It is classified as a hydrophilic. The ring structure puts restrictions on the allowed configurations when it is incorporated in proteins. It creates a bent or kink in the protein backbone structure.
- Acidic amino acids with an acidic side chain are hydrophilic. These amino acids has carboxylic acid (\(\ce{-COOH}\)) group in their side chain that ionize to anion \(\ce{-COO^{-}}\) under physiological conditions.
- Basic amino acids with a basic side chain are hydrophilic. These amino acids have basic primary or secondary amine groups in their side chain that ionize to cation \(\ce{-NH3^{+}}\) or \(\ce{-NRH2^{+}}\). Histidine has an amine group that has pKa 6.04 and is not ionized at pH 7.4, but it is placed in this group as it ionizes at pH below 6.0.
Acid-base nature of \(\alpha\)-amino acids
The pKa is a measure of the strength of an acid, i.e., the lower the pK a stronger the acid. Amino acids have \(\ce{-COOH}\) group that is acidic with pK a 2-3 and \(\ce{-NH2}\) on adjacent \(\ce{C}\) that is basic with pK a ~40. Conjugate acid of \(\ce{-NH2}\), i.e., \(\ce{-NH3^{+}}\) has pK a ~10. Acids lose their proton when they are in a medium with a pH higher than the pK a of the acid. For example \(\ce{-COOH}\) exist as \(\ce{-COO^{-}}\) in physiological medium with pH ~7.4. Bases gain protons in a medium with a pH lower than the pK a of their conjugate acid. For example, \(\ce{-NH2}\) exist as \(\ce{-NH3^{+}}\) in physiological medium with pH ~7.4. Amino acids exist as zwitterions, i.e., have both cation \(\ce{-NH3^{+}}\) and anion \(\ce{-COO^{-}}\) in the same molecule. Some amino acids have an additional acid or base group in their side chain that also ionizes depending on the pH of the medium.
The gain or loss of protons is an equilibrium process. In a strongly acidic medium, basic groups gain more protons than the protons lost by their acid groups. So, the amino acids have an overall positive charge in a strong acid medium. In a strongly basic medium basic groups gain fewer protons than those lost by their acid groups. So, the amino acids have an overall negative charge in a strongly basic medium. At a certain pH in the middle, an amino acid has an equal positive and negative charge and is neutral overall.
An amino acid's i soelectric point (pI) is the pH at which it has equal positive and negative charges and carries no net charge, i.e., it is neutral overall.
The pK a value of \(\ce{-COOH}\) and \(\ce{-NH3^{+}}\) on the \(\alpha\)-\(\ce{C}\), pK a values of acidic or basic groups in the side chain, and pI values of 20 \(\alpha\)-amino acids found in proteins are listed in Table 1 below.
| Amino acid | Three letters abbreviations | One letter abbreviations | pK a of \(\alpha\)\(\ce{-COOH}\) | pK a of \(\alpha\)\(\ce{-NH3^{+}}\) | pK a of side chain group | Isoelectric point (pI) |
|---|---|---|---|---|---|---|
| Alanine | Ala | A | 2.34 | 9.69 | – | 6.00 |
| Arginine | Arg | R | 2.17 | 9.04 | 12.48 | 10.76 |
| Asparagine | Asn | N | 2.02 | 8.80 | – | 5.41 |
| Aspartic acid | Asp | D | 1.88 | 9.60 | 3.65 | 2.77 |
| Cysteine | Cys | C | 1.96 | 10.28 | 8.18 | 5.07 |
| Glutamic acid | Glu | E | 2.19 | 9.67 | 4.25 | 3.22 |
| Glutamine | Gln | Q | 2.17 | 9.13 | – | 5.65 |
| Glycine | Gly | G | 2.34 | 9.60 | – | 5.97 |
| Histidine | His | H | 1.82 | 9.17 | 6.00 | 7.59 |
| Isoleucine | Ile | I | 2.36 | 9.60 | – | 6.02 |
| Leucine | Leu | L | 2.36 | 9.60 | – | 5.98 |
| Lysine | Lys | K | 2.18 | 8.95 | 10.53 | 9.74 |
| Methionine | Met | M | 2.28 | 9.21 | – | 5.74 |
| Phenylalanine | Phe | F | 1.83 | 9.13 | – | 5.48 |
| Proline | Pro | P | 1.99 | 10.60 | – | 6.30 |
| Serine | Ser | S | 2.21 | 9.15 | – | 5.68 |
| Threonine | Thr | T | 2.09 | 9.10 | – | 5.60 |
| Tryptophan | Trp | W | 2.83 | 9.39 | – | 5.89 |
| Tyrosine | Tyr | Y | 2.20 | 9.11 | 10.07 | 5.66 |
| Valine | Val | V | 2.32 | 9.62 | – | 5.96 |
Essential amino acids
Nine amino acids are essential because humans can not synthesize them fast enough to meet their demands.
The essential amino acids are valine, isoleucine, leucine, methionine, phenylalanine, tryptophan, threonine, histidine, and lysine.
The essential amino acids are obtained from foods. Foods from animal sources, e.g., eggs, milk, fish, meat, etc., are complete foods with all the essential amino acids. Foods from plant sources, e.g., wheat, rice, corn, etc., are usually deficient in one or more essential amino acids. So, vegetarians have to eat various vegetarian foods to obtain all the essential amino acids.
- Wheat, rice, and oats are deficient in lysine.
- Corn is deficient in lysine and tryptophan.
- Soy is deficient in methionine.
- Beans are deficient in methionine and tryptophan.
- Peas and peanuts are deficient in methionine
- Almonds and walnuts are deficient in lysine, tryptophan
- Foods from animal sources, e.g., milk, eggs, meat, fish, etc., have all the essential amino acids.